Solar Magnetism in the Polar Regions
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- Petrie, G.J.D. Living Rev. Sol. Phys. (2015) 12: 5. doi:10.1007/lrsp-2015-5
This review describes observations of the polar magnetic fields, models for the cyclical formation and decay of these fields, and evidence of their great influence in the solar atmosphere. The polar field distribution dominates the global structure of the corona over most of the solar cycle, supplies the bulk of the interplanetary magnetic field via the polar coronal holes, and is believed to provide the seed for the creation of the activity cycle that follows. A broad observational knowledge and theoretical understanding of the polar fields is therefore an essential step towards a global view of solar and heliospheric magnetic fields. Analyses of both high-resolution and long-term synoptic observations of the polar fields are summarized. Models of global flux transport are reviewed, from the initial phenomenological and kinematic models of Babcock and Leighton to present-day attempts to produce time-dependent maps of the surface magnetic field and to explain polar field variations, including the weakness of the cycle 23 polar fields. The relevance of the polar fields to solar physics extends far beyond the surface layers from which the magnetic field measurements usually derive. As well as discussing the polar fields’ role in the interior as seed fields for new solar cycles, the review follows their influence outward to the corona and heliosphere. The global coronal magnetic structure is determined by the surface magnetic flux distribution, and is dominated on large scales by the polar fields. We discuss the observed effects of the polar fields on the coronal hole structure, and the solar wind and ejections that travel through the atmosphere. The review concludes by identifying gaps in our knowledge, and by pointing out possible future sources of improved observational information and theoretical understanding of these fields.
KeywordsPolar magnetic fieldsPhotosphereChromosphereFlux transportCoronal holesSolar windProminencesCoronal mass ejections (CMEs)
The magnetic field located at the heliographic poles of the Sun has a large-scale (±60–90° latitude) unipolar distribution. These distributions have opposite polarity at the two poles, except during times of polarity reversal. Although the polar field is highly structured, with small-scale features of facular (kG) strength, the average flux density of the polar fields is only about 5 G. The polar fields are therefore much weaker than active region fields, and they contain less magnetic flux than a major active region. Nevertheless, they have far-reaching importance because of their unipolarity over large spatial scales, and because of their role in the solar activity cycle.
Most polar magnetic flux does not connect back to the Sun, unlike active region flux which is generally closed. This means that the polar fields supply most of the interplanetary mean field and channel most of the fast solar wind. The open polar flux takes the form of polar coronal holes, which dominate the large-scale structure of the corona over most of the cycle. The exception is when the polar fields are reversing polarity, which occurs approximately every 11 years during solar activity maximum. This observed interrelationship between polar field reversal and the solar activity cycle is believed to be just the surface manifestation of a unified cycle linking the active regions and polar fields. This review will summarize in some detail the basic observed properties of the polar fields, their role in the solar cycle according to observations and models, and their influence over coronal and heliospheric phenomena.
The importance of the polar fields to the global magnetic field of the Sun therefore lies both in their central role in the solar cycle and in their dominant influence over the heliosphere. Frustratingly, the polar fields are the most difficult of the Sun’s surface fields to measure. The polar fields are intrinsically weak compared to the active region fields located at low latitudes, and there is a large projection angle when the polar latitudes are observed from Earth. For these reasons, measurement of the polar fields is challenging, particularly with the spatial resolution of present-day full-disk magnetographs.
Routine and continuous full-disk line-of-sight magnetogram observations began at Mt. Wilson Observatory (MWO) in the 1960s (Howard, 1989), and have been taken at the National Solar Observatory (NSO) and the U. Stanford’s Wilcox Solar Observatory (WSO) since the 1970s (Livingston et al., 1976; Svalgaard et al., 1978). Regrettably, the MWO observations stopped in 2012, but the data series from NSO and WSO continue to the present day. Only since the launch of the Hinode spacecraft in late 2006 have detailed vector maps of the polar magnetic field distribution been possible (Tsuneta et al., 2008), under optimal observing conditions, viz. when the pole is tilted a little (about 7.25°) towards us. In this review we will describe the Hinode results in some detail, before relating them to the more long-term results based on synoptic observations of the line-of-sight field component. These observations revealed a unipolar but highly complex and non-uniform flux distribution, containing ubiquitous fields of greater strength (> 1 kG) than many had previously expected, though bundles of kilogauss field had previously been observed at high latitudes (Homann et al., 1997; Okunev and Kneer, 2004; Blanco Rodríguez et al., 2007) as well as in the low-latitude quiet-Sun photosphere (Orozco Suárez et al., 2007).
From full-disk measurements of the line-of-sight field, butterfly diagrams (latitude-time plots of the inferred radial field distribution) can be constructed. These plots are very useful because they reveal the interactions between strong, low-latitude fields and weak, high-latitude fields including the polar fields, that occur over long timescales. Babcock (1959) observed the asymmetric pattern of the cycle 19 polar reversal, the first observations of reversing polar fields, and reported that the south polar reversal preceded the north by nearly 18 months. Guided by some earlier pioneering work in solar dynamo theory, Babcock (1961) presented his phenomenological model for the solar cycle, based on his full-disk magnetogram observations. This model was supported by a comparison of polar facular counts and sunspot numbers by Sheeley Jr (1964), and a numerical kinematic flux transport model by Leighton (1969). This cyclical interaction between the active regions and polar fields has been a central focus of observational and theoretical study for solar physicists since these initial studies. We will discuss the observed relationship in some detail before reviewing numerical kinematic flux transport models for the cycle, beginning with Leighton (1964, 1969) and continuing to the present.
Of course, kinematic models for the transport of photospheric flux are unable to describe the physics of the polar fields in their full complexity, but they have given us critical insight into the causes of polar field phenomena. The effectiveness of these models has improved. Under the guidance of ever more refined observations of the photosphere and the interior, the models have become steadily more flexible, stable and accurate. It is now possible to produce full-surface snapshots of the photospheric field using these models, and these “synchronic” synoptic magnetograms (sometimes referred to synoptic charts or maps) are an essential raw material for models of the solar atmosphere.
Whereas the dynamics of the interior and photosphere are dominated by the fluid flow, the much less dense plasma in the corona (up to about a solar radius) is dominated by the magnetic field, whose distribution and structure are determined by the surface magnetic flux distribution. Models of the atmospheric field show the dominant influence of the polar fields via the axial dipole component, and observations of coronal holes, solar wind distributions, and prominence eruptions and coronal mass ejections, the solar phenomena that most directly impact us here on Earth, all bear the mark of the waxing and waning influence of the polar fields over the cycle.
Observations of polar faculae and filaments have been taken over many decades, starting much earlier than full-disk magnetograph measurements, and they enable statistical comparisons of cycle-by-cycle polar fields and sunspot numbers (Sheeley Jr, 1964; Muñoz-Jaramillo et al., 2013). Filaments mark neutral lines between predominantly unipolar bodies of weak, opposite-polarity flux at high latitudes, and they enable us to follow the progress of the poleward-transported flux that forms the polar fields. Their eruptions at high latitudes, and the removal of the helicity that they have carried there from lower latitudes, are an essential part of the polar field reversal. We will summarize observational and modeling results concerning the role of filaments and eruptions in polar magnetism.
Petrie et al. (2014) recently reviewed observations and models of the interactions between active regions and the polar field, focusing in particular on the interrelated phenomena that are observed to migrate in both directions across the high-latitude corridor between the active and polar latitudes, and Petrie and Ettinger (2015) reviewed in detail the interaction of decayed active-region flux with polar fields via poleward surges at the photospheric level. While there is overlap between that review and this one, here we focus much more closely on observations and flux transport modeling of the polar fields, before reviewing atmospheric phenomena that clearly exhibit signs of the polar fields’ global influence.
The review is structured in three broadly-themed sections, designed to convey the basic observed facts and theoretical understanding of the polar fields, before discussing their far-reaching influence and importance throughout the solar atmosphere. Section 2 presents observational analyses of direct polar field observations, beginning with the high-resolution photospheric vector measurements of Hinode, relating them to the more traditional line-of-sight photospheric measurements, and introducing new types of synoptic data products, before summarizing the patterns of flux transport in the 40 years of magnetogram observations from NSO. Section 3 continues the theme of flux transport and focuses on it in the context of modeling, beginning with the initial phenomenological and kinematic models of Babcock (1961) and Leighton (1964, 1969), and continuing with recent efforts to explain the unusual behavior of the polar fields during the cycle 23 minimum. Section 4 explores the global influence of the polar fields over the heliospheric structure, and the solar wind and ejections that travel through it. We will conclude in Section 5.
2 Observations of the Polar Magnetic Field
2.1 High-resolution observations of polar fields
While the polar regions are very important to several of the major branches of global solar physics, from the solar dynamo to the acceleration of the fast solar wind, the magnetic behavior of the polar fields is not comprehensively understood. Polar field measurements are very challenging. The tilt angle between the solar rotation axis and our viewpoint on the Earth’s ecliptic plane, usually referred to as the B0 tilt angle, is approximately 7.25°, meaning that the viewing angle of the poles is never less than about 83°. Optimal polar viewing angles only occur annually, on 6 March for the south pole and 8 September for the north pole. Over the first/second half of each year the north/south pole is unobservable from the direction of Earth. When a pole is observable, strong intensity gradients and foreshortening at the limb, and variable seeing conditions in the case of ground-based observations, all pose difficulties.
Full Stokes polarimetry for the polar regions has been performed only rarely and, until the Hinode satellite was launched in late 2006, such efforts were generally confined to ground-based observations under variable seeing conditions. Also, polar vector field observations have often been restricted to limited fields of view and therefore haven’t provided a picture of the global distribution of the polar field.
The estimated total polar flux according to these measurements (above 70°) was 5.6 × 1021 Mx with the nominal filling factor applied and 2.5 × 1022 Mx with filling factor set to 1. Thus the total polar magnetic flux was less than that of a major active region. If the polar fields were evenly distributed, they would have average strength 3.1 G with the nominal filling factor applied, 13.9 G with filling factor set to 1, and 10.0 G with 50% of stray light taken into account (Tsuneta et al., 2008). This estimate of average field strength is roughly consistent with the values derived from lower-resolution synoptic line-of-sight measurements that we will discuss in Section 2.7.
The patches at the higher end of this range were an order of magnitude larger than those found in the quiet Sun, and were nearly as large as pores. The number densities of flux concentrations as functions of total magnetic flux were found to decrease over the four orders of magnitude studied. The polar regions had comparable number densities of both polarities in the flux range 1015—1017 Mx but significant flux imbalances in the patches larger than 1017 Mx. In contrast the quiet Sun’s flux concentrations were flux-balanced over their entire flux range.
As in the study by Tsuneta et al. (2008), two distinct populations were found in the polar regions, large concentrations that varied with the solar cycle and determined the overall polarities of the polar fields, and smaller concentrations of mixed polarity that appeared to be cycle-invariant. Almost all large patches (> 1017 Mx) had the same polarity at each pole while the population of smaller patches had approximately balanced flux. The polarities of the polar caps were therefore determined by the large patches.
Comparing Figures 2 and 4, we see that the largest and strongest flux concentrations generally have approximately vertically-directed field, but that horizontal field intensity is significantly greater than the vertical field intensity on average. Also, the vertical field in large concentrations has more long-term variability than the other fields, consistent with the idea that it is the vertical field that determines the overall polarity of the polar caps and that plays a role in the global behavior of the solar field and the solar cycle. The horizontal fields, despite their larger average strength, appear to have much more limited influence, connecting small, adjacent magnetic features that do not contribute significantly to the large-scale polar flux and do not play a major role in the global cycle.
2.2 Are the polar fields radial, and do they have a topknot distribution?
Moreover, longitudinal observations, though they can have sub-gauss instrumental noise levels, suffer from the foreshortening problem associated with our large viewing angle from the ecliptic plane, and the fact that any radially-directed field at polar latitudes, such as those identified in the largest concentrations of flux at the poles by Hinode, have only a small component directed along our line of sight. In spite of these problems, it is possible to extract much information about the solar surface fields’ tilt angles from long time series of longitudinal field measurements by using changes in the viewing angle caused by solar rotation and the tilt of the rotation axis with respect to the ecliptic plane.
The conclusion of Svalgaard et al. (1978) that much of the photospheric field is approximately radially directed has had a major influence on solar global atmospheric modeling, as we shall see in Section 2.5, but it has not escaped criticism. Rudenko (2004) argued that a similar linear decrease could be produced by a change of sign of line-of-sight flux near the limb for a non-radial field vector, and therefore concluded that this type of calculation could not determine whether or not the photospheric field is radial.
Petrie and Patrikeeva (2009) repeated the experiment using photospheric and chromospheric magnetogram images from the SOLIS/VSM, first by binning the data using as the selection criterion the sign of the line-of-sight field component at central meridian as did Svalgaard et al. (1978), and second by binning positive and negative line-of-sight fields separately at every position on the disk. The latter experiment had the unsettling property that different measurements of the field at the same location on the photosphere appeared in different sides of the rhombus-like plot, but this exercise did enable the authors to demonstrate the problem flagged by Rudenko (2004). Whereas the calculation performed using the method of Svalgaard et al. (1978) yielded rhombus-like plots for both photospheric and chromospheric fields, the second experiment resulted in contrasting plots for the two sets of observations: the photospheric plot was an almost unchanged rhombus-shaped graph while the chromospheric plot had much-altered, nearly constant graphs. This plot (not shown) contrasted the photospheric and chromospheric fields, suggesting that the former are nearly radial and the latter are not.
When Petrie and Patrikeeva (2009) applied this method to SOLIS Ca ii 6542 A chromospheric field measurements, these fields had evolved too much to produce well-defined line-of-sight flux functions of latitude, except during 2008 when the north polar field was approximately radial and the south polar field was expanding towards the observer, i.e., super-radially (see also Jin et al., 2013).
In an earlier study by Raouafi et al. (2007) of polar field measurements in the same series of SOLIS Ca ii 6542 Å chromospheric magnetograms, the analysis focused on small-scale features in the polar field structure. They found that the number density and line-of-sight magnetic flux of identified magnetic elements decreased poleward as functions of latitude. The superficial disagreement between this result and the results of Petrie and Patrikeeva (2009) may be due to the effects of foreshortening on the visibility of field structure: the effective spatial resolution in heliographic coordinates decreases sharply close to the limb, and consequently less surface structure, such as the magnetic elements studied by Raouafi et al. (2007), can be identified as one observes closer to the limb.
2.3 Vector photospheric synoptic maps
Conspicuously absent from this vector map are measurements from the polar latitudes. This is because the high-latitude fields are too weak to be reliably inverted from Stokes data with the 1″ pixel size of the SOLIS/VSM images. The effective spatial resolution at the poles is also compromised by foreshortening effects and atmospheric by seeing. This exercise demonstrates the challenging nature of routine synoptic vector field measurements at high latitudes, where the field is weak. The Hinode SOT has demonstrated that vector field at the poles can be measured in great detail under optimal conditions, but it remains true that routine synoptic measurements of the polar magnetic vector have not yet been achieved. SOLIS and HMI lack the resolution or sensitivity to provide detailed maps of the polar vector field. Synoptic line-of-sight magnetograms from, e.g., SOLIS, HMI, NSO’s Global Oscillations Network Group (GONG) and WSO include measurements of weak fields down to 1 G or weaker, much less than the average flux density of the polar field measured by Hinode as described in Section 2.1, but enough to catch the approximately 5 G large-scale high-latitude field distribution. For this reason, the weak line-of-sight field can be diagnosed at high latitudes with low-resolution instruments.
More recently, SOLIS has performed long-exposure observations of polar latitudes with more limited fields of view. Also it is planned to combine vector field measurements of strong fields and line-of-sight measurements of weak fields in a single map. As Section 2.1 indicates, full-Stokes measurements will play a central role in characterizing the polar fields in the future, and over time the contribution of routine synoptic measurements to this process will increase.
2.4 Polar field interpolation for missing data
As we have seen, the photospheric magnetic field at high latitudes, in particular at the poles, is difficult to observe from Earth because the angle between the solar rotation axis and the ecliptic plane is small, 7.25°. So far all of our solar magnetographs have been confined to the ecliptic plane. The Ulysses spacecraft traveled in a poloidal orbit around the Sun but it did not carry a magnetograph. From Earth or locations nearby, the south pole is visible early in the year and the north pole late in the year, with optimal viewing angles on 6 March and 8 September, respectively. For most of the year, one or the other pole is not observable. At all times the large projection angle makes it difficult to resolve magnetic features at the poles using a present-day full-disk synoptic magnetograph. An observation of the polar field by such a magnetograph, such as SOLIS or HMI, generally shows a less structured, almost unipolar flux distribution covering the polar cap, whose average field strength peaks somewhere in the range 5–10 G at solar minimum. Measurements of even a weak line-of-sight field component are often reliable to high latitudes but, as described in Section 2.2, the properties of the measured fields are not well defined all the way to the limb, reflecting the fact that the measurements are not reliable there. Pixels at the edges of full-disk magnetograms tend to be noisier than those near disk-center and, because the large-scale fields are generally approximately radially directed, the line-of-sight component of the field near the limb tend to be weaker than those near disk-center. The limb field data are therefore more prone to being unreliable. The limb data corresponding to low-latitude locations can be compared to observations taken at other times when these locations are facing the Earth, but this solution is not available for high-latitude fields.
In tension with this unfortunate fact, there are several important branches of solar physics that rely on accurate descriptions of the polar field distribution. For example, the large-scale distribution of the polar field has a dominant influence on the structure of global coronal models, as well as the models for the solar wind based on them (Section 4.3). Furthermore, modelers of the global solar dynamo (Section 3) rely on measured polar field strengths to build and test their models.
In the absence of optimal conditions for measuring the polar fields, field data for these latitudes must be derived either by flux-transport modeling, where the polar field is built from well-measured active-region fields that are transported to the poles as we will discuss in Section 3, or by interpolating or extrapolating from better-quality measurements of lower-latitude fields. For many years solar observatories have used simple interpolation or extrapolation techniques to fill missing pixels in their synoptic maps.
One way to mitigate the problem is to exploit high-latitude observations taken with optimal B0 tilt angles. The large-scale distribution of the polar fields evolves gradually enough that it is possible to derive a reasonable field-strength estimate by interpolating between annual measurements taken when the pole in question was tilted towards the Earth. Even with this information some spatial interpolation is always necessary. Spatial interpolation across the pole based on lower-latitude measurements can be performed in one dimension by fitting curves in meridional slices, or in two dimensions by fitting surfaces.
A successful pole-fitting method must give a reasonable estimate for the polar fields by using good-quality observations to the fullest possible extent. There is no obviously optimal solution to this problem but some comparative studies of various methods have been made. Liu et al. (2007) compared the results of seven different pole-filling techniques. These included one-dimensional cubic spline interpolation with and without smoothing, two-dimensional low-degree polynomial surface fitting with or without temporal interpolation, a “topknot” model based on fitting a surface of the form Bp cosnθ (Svalgaard et al., 1978, see Section 2.2), and the flux transport method described by Schrijver (2001) (see Section 3.5). Liu et al. (2007) concluded that the best technique for filling missing polar data was one combining two-dimensional low-degree polynomial surface fitting and temporal interpolation.
2.5 The radial photospheric field and potential coronal field models
One major application of full-surface solar magnetic field measurements is as boundary data for models of the solar atmosphere. These models are of practical as well as scientific importance because they help us to locate sources of the solar wind. Global models for the solar atmosphere place a heavy burden on the polar field measurements because, while the polar fields are difficult to measure, they play a leading role in structuring these global models. In this section and in Section 2.6 we discuss efforts to apply line-of-sight observations from the photosphere and chromosphere as boundary data for global models.
A sensitive issue with the use of photospheric data as boundary data in atmospheric modeling in general is the great physical mismatch between the highly forced, plasma-dominated photospheric layers where the measurements originate, and the atmospheric models which either are force-free or, in the MHD case, do not resolve the extremely complex lower atmospheric layers between the photosphere and the corona. In this sense, the boundary data supplied by the photospheric field measurements are physically inconsistent with the models.
The photospheric field is non-potential, being in a fluid-dominated medium, and is typically nearly radial at the height of the measurements, as Section 2.2 showed. Matching the line-of-sight component of a potential field model for the corona to line-of-sight measurements of the photospheric field is strictly not a physically consistent approach because the field is not likely to be approximately current-free at the height of the observations. Wang and Sheeley Jr (1992) argued that a procedure more consistent with nearly-radial, non-potential photospheric fields is to convert the line-of-sight field measurements to radial fields by dividing by cos ρ, where ρ is the heliocentric angle between the line-of-sight vector and the local vertical. This method generally implies a discontinuity between the vanishing horizontal field component in the photosphere and the finite horizontal field component of the model immediately above the photosphere. Wang and Sheeley Jr (1992) interpreted this discontinuity as a mathematical idealization of a very thin but finite boundary layer between the non-potential and approximately radial photospheric field and the approximately potential and non-radial coronal field. On global coronal scales this model estimates the radial magnetic flux into the corona as well as can be done based on line-of-sight observations. It clearly enhances the strength and influence of fields observed near the limb, most notably the polar fields.
The radial-field approach generally produces open-field and source-surface neutral line distributions that match observed coronal holes and streamer structures better than does the line-of-sight approach. Wang and Sheeley Jr (1992) argued that the radial field approximation makes the best possible use of the available magnetogram data, consistent with the conclusion of Section 2.2 that the photospheric field is typically nearly radial at the height of the measurements.
2.6 Chromospheric synoptic maps and potential-field extrapolations
Most global coronal modeling relies on photospheric radial field maps based on line-of-sight measurements, implicitly applying the physical assumptions discussed by Wang and Sheeley Jr (1992) described in the last section. This is partly due to the much wider availability of photospheric line-of-sight field measurements than measurements from higher in the solar atmosphere, though routine chromospheric full-disk measurements and synoptic maps are available from the SOLIS/VSM. In this section we describe a recent effort to apply chromospheric field observations as lower-boundary data for coronal PFSS models.
Chromospheric fields differ significantly from the underlying photospheric fields. Whereas photospheric fields are directed approximately radially over most of the solar surface, the chromospheric fields often form canopy structures so that the fields spread horizontally (Jones, 1985), as discussed in Section 2.2. A powerful motivation for using chromospheric field observations in coronal modeling is that the chromosphere is not separated from the corona by a complex transition region like the photosphere is, and the chromosphere is much more similar to the corona physically, generally being magnetically dominated and approximately force-free. The main challenge in using chromospheric data is that, since the field expands in all directions the radial magnetic flux into the atmosphere cannot be easily estimated from line-of-sight chromospheric data — there is no analog of the radial field assumption for the chromosphere, as also discussed in Section 2.2. Of course one could apply chromospheric line-of-sight data directly as boundary data to the PFSS model, relying on the model to determine the radial flux, but this turns out to be an unreliable method. A major difficulty is that the global radial flux generally does not balance because the high-latitude radial flux is not estimated accurately by a potential field model constrained by line-of-sight observations. PFSS models are constructed neglecting the monopole component of the surface magnetogram. A large monopole component in the surface data, when subtracted for modeling purposes, is often indicated by a spurious net displacement of the neutral line at the source surface (outer boundary) from the equator. Such artifacts are characteristic of models whose high-latitude radial fluxes are not accurately represented. If the potential field model does not reproduce the real tilt of the chromospheric polar fields then the polar flux is doomed to be poorly estimated.
It is much more practical to work with boundary data for the radial field component. Jin et al. (2013) took such an approach. They developed a method for producing synoptic maps for the chromospheric radial field component based on full-disk chromospheric Ca ii line-of-sight magnetograms taken by the SOLIS/VSM between April 2006 and November 2009. They used the annual change in our viewing angle from Earth of the polar regions to estimate the radial and meridional components of the chromospheric polar fields. Significant radial and meridional components were detected at both poles, and the south polar field was tilted more strongly away from the pole (see also Section 2.2, Petrie and Patrikeeva, 2009).
Because of loss of resolution due to foreshortening and the nearly radial orientation of the vector field, line-of-sight field components observed near the limb appear weaker in general than line-of-sight fields observed near the center of the disk. For nearly unipolar fields, such as the polar fields, the loss of resolution is not necessarily a problem because not much flux is likely to be lost. The center-to-limb variation of polar field measurements cannot be investigated directly because we can only observe the polar fields from the ecliptic plane with large viewing angles. Instead, Jin et al. (2013) investigated the center-to-limb variation of nearly unipolar low-latitude regions, which are believed to be similar to polar fields. They tracked 20 unipolar regions, of unsigned flux density > 8 G and ratio of major-polarity flux to minor-polarity flux > 3. They arrived at a correction function increasing from 1 at central meridian to about 2.25 at the limb. They corrected their full-disk measurements using this function.
Thus, it is possible to estimate the polar flux from chromospheric measurements after some work, but the photospheric radial field approximation provides a simpler and approximately equally accurate estimate. For this reason, PFSS and MHD models extrapolated from boundary data based on photospheric line-of-sight measurements and the radial field assumption remain competitive (Section 2.5).
2.7 Cycle relationship between polar and active fields: the magnetic butterfly diagram
An important advantage of studying approximately radial fields is that the full magnetic flux through the photosphere can be estimated reasonably accurately over most of the solar disk. However, because the solar rotation axis is tilted at an angle of 7.25° with respect to the ecliptic plane, the fields near the solar poles are either observed with very large viewing angles or, for six months at a time, not observed at all. Also the noise level is inflated near the poles by the radial field correction discussed above. For these reasons we have had to fill the locations in the butterfly diagram nearest the poles using estimated values for these fields. This is a well-known problem in the construction of synoptic magnetograms (e.g., Sun et al., 2011, see Section 2.4). The polar flux is generally found to become stronger as one observes further poleward (Svalgaard et al., 1978; Wang and Sheeley Jr, 1988; Petrie and Patrikeeva, 2009, see Section 2.2), and the polar field distribution in Figure 17 reflects this. The fields at the highest latitudes in Figure 17 were calculated from measurements taken with advantageous solar axis tilt (B0) angles (B0 > 5° for northern and B0 < −5° for southern high-latitude fields). The resulting fields are well defined and regular functions of time for all but the two most poleward sin(latitude) bins at each pole (there are 180 uniformly-spaced sin(latitude) bins in the butterfly diagram overall). For these two most poleward bins, simulated data were used based on a polynomial fit for each image. The simulated data were then blended with the measurements in a B0-dependent fashion.
Figure 17 covers cycles 21–23 in their entirety, as well as the end of cycle 20 and the ascent and maximum of cycle 24. The diagram shows several distinctive patterns in the long-term behavior of the fields at active and polar latitudes, and at latitudes in between. The active fields begin each cycle emerging at latitudes around ±30° and subsequently emerge at progressively lower latitudes on average, a phenomenon referred to as Spörer’s law, creating the distinctive wings of the butterfly patterns first reported by Maunder (1913). The diagram also shows the change of polarity of the polar fields once each cycle, coinciding with activity maximum but not occurring simultaneously at the two poles. Between the active and polar latitudes, around ±50°, there is clear evidence of poleward flux transport of both polarities in each hemisphere during each cycle, which appears most intense during the most active phases of the cycle. Most of the flux that emerges in active regions cancels with flux of opposite polarity, but a critical proportion survives as weak flux that is carried poleward by the a poleward surface meridional flow and by the diffusion-like effect of the photospheric convection (see Section 3). This poleward drift of the weak, decayed magnetic flux appears in Figure 17 as plumes of one dominant polarity, the trailing sunspot polarity, at high latitudes between about 40° and 65°, sometimes referred to as a “rush to the poles”. Such patterns were first reported by Bumba and Howard (1969) who named them unipolar magnetic regions. The dominant polarity of the poleward plumes is of opposite sign in each hemisphere, and it alternates from cycle to cycle. Howe et al. (2013) compared the poleward migration rate estimated from the high-latitude surges in a Kitt Peak magnetic butterfly diagram to the subsurface meridional flow rates measured in helioseismic data from the GONG network since 2001, and found the two rates to be in reasonable agreement. Note that there is no evidence of equatorward high-latitude counter-cell meridional flow in Figure 17. The high-latitude flux transport appears to be exclusively poleward.
In these “Babcock-Leighton” models, the cycle-dependent polarity patterns and the Joy’s law tilt preference are assumed to be responsible for creating the polar field cycle from the activity cycle. The contribution to the polar field in a given hemisphere from decayed active region flux seems to be proportional to the total active region flux in that hemisphere and to the latitude displacement between the positive and negative active-region flux centroids in that hemisphere. Petrie (2012) found a correlation between the annual-averaged high-latitude (approx. ±50°) poleward stream fluxes and the product of the annual-averaged latitude flux centroid displacements and total active-region fluxes. A correlation was also found between the annual-averaged high-latitude poleward stream fluxes and the annual-average polar (±63–70°) field changes. These correlations were found in both hemispheres with both NSO Kitt Peak data and Mt. Wilson Observatory (MWO) 150-foot tower data.
The Joy’s law tilt and polarity biases are not strict. Even in averaged form, the poleward flux surges appear to be approximately periodic in time (Ulrich and Tran, 2013) with frequent changes in sign. The widths of the surges are approximately equal to the meridional flow travel time between the active latitudes and the poles, between 1 and 1.5 yr, also approximately equal to the equatorial dipole decay time (Wang et al., 2009). The streams originate from sizable densely packed groups of sunspots that survive for many months and are almost continuously refreshed during their lifetimes by the emergence of new bipoles (Gaizauskas et al., 1983; Schrijver and Zwaan, 2000). The interaction of multiple tilted bipoles may produce large poleward surges of flux and large polar field changes in a relatively short time.
Figure 18 shows that, during the decline of cycle 23, the centroids converged in both hemispheres around 2003, and stayed together until the end of the cycle — see also Section 3.6. In the south this convergence occurred earlier, around 2001. Correspondingly, according to Figure 19, the south polar field became stable after 2001 and the south polar field around 2003. There is evidence in Figure 18 that this was not the first time the positive and negative latitude centroids converged, During cycle 22 the centroid separation and the activity amplitude (measured in terms of active region flux) peaked together in the north, compatible with the abrupt polar field reversal, followed by convergence of the latitude centroids and stability in the polar fields, shown in Figure 19. In the south the centroid separation was small during the ascent of cycle 22, but increased as the decline of the cycle got under way. Meanwhile, the south polar field began to reverse in the south more slowly than in the north, but sped up and reversed as the centroid separation increased.
When cycle 24 began in 2010, the northern hemisphere developed ahead of the south. Figure 17 shows that positive field-carrying surges traveled poleward as shown by the red streaks, causing the north polar field to reverse first as shown in Figures 17 and 19. This is consistent with the Hinode results of Shiota et al. (2012) shown in Figure 4, in which the polar flux density decreased in the north than in the south. Svalgaard and Kamide (2013) and Mordvinov and Yazev (2014) emphasized the link of this asymmetry to an asymmetry in the decayed active region fields that are transported poleward. This asymmetry is a major reason for the asynchronous north and south polar reversals seen in Figure 19. During 2013, however, a surge of negative flux moved poleward, threatening to cancel the progress of the north polar field. In the south the opposite happened: the activity started late and initially sent mixed polarity-flux poleward, and the south polar field was slow to reverse. But the south polar field has been fed a steady diet of negative flux in the past few years, represented by the blue streaks in the bottom right of Figure 17. In response the south polar field has reversed and has been strengthening ever since. The most recent active regions appear to be sending more negative flux towards the south pole, followed by a more weakly net-positive surge beginning early this year (2015) but a long-duration positive surge towards the north pole. The fate of the north polar field reversal is therefore becoming clearer after its interrupted reversal. Although at the time of writing the north polar field has turned positive again and is strengthening, we still have highly asymmetric polar fields.
3 Polar-Field Reconstruction in Kinematic Flux-Transport Models for the Solar Cycle
In the previous section, we summarized observations of high-latitude magnetic fields, special observations at high resolution and long-term synoptic observations. The synoptic observations revealed a coherent pattern of high-latitude flux transport, a “march to the poles” of surface magnetic flux. In this section, we will begin in Section 3.1 by discussing the phenomenological model of Babcock (1961), the first attempt to unite the diverse cyclical patterns of bipolar active regions and polar fields into a coherent picture of the solar cycle. We will then describe in the following subsections the subsequent numerical kinematic modeling of the cycle, an effort that continues to this day.
Models of the solar magnetic cycle kinematically have focused on the role of global plasma flows in producing the organized behavior of the fields. Kinematic flux transport models for the solar cycle are governed by the induction equation relating the fluid flow and the magnetic flux transport. In the axisymmetric case, the induction equation breaks down into two simple coupled equations, one for the vector potential of the poloidal field and the other for the toroidal field component — see, e.g., Charbonneau (2010), Eqs. (11, 12). In the equation for the toroidal field, the conversion of poloidal field to toroidal field is clearly represented by a source term describing the shearing effect of interior differential rotation. However, the equation for the poloidal field has only an advection term that can neither create nor destroy magnetic flux, and a diffusion term that can only destroy magnetic flux. Thus whatever an axisymmetric flow field does, the poloidal field must ultimately decay away, and the toroidal field must also decay as a consequence (Cowling, 1933).
Therefore, a non-axisymmetric process is needed to provide a poloidal field source. Parker (1955) introduced a poloidal field source term representing the effect of helical turbulent flows on buoyantly rising loops that twist in response to the Coriolis force, creating poloidal field components in previously toroidal loops. Such a process, commonly called the “alpha effect”, and its associated poloidal field source term, enables dynamo models to produce cyclical behavior. In the “Babcock-Leighton” mechanism (Babcock, 1961; Leighton, 1969) the alpha effect is contained in the Joy’s law tilt of the active regions, whose flux is converted into polar flux by surface and near-surface transport.
Flux transport dynamo models have been the main theoretical tool for understanding the origins of the solar cycle and solar activity, and have produced a complex theoretical picture of how the solar dynamo may operate and how the solar convection zone and tachocline may behave. For simplicity, we will focus here on Babcock-Leighton-type models, both data-driven and self-excited, where near-surface flows produce the alpha effect. The interested reader is referred to Charbonneau (2010) for a much more general discussion of solar dynamo models.
In both, surface flux transport models and self-excited flux transport dynamo models, the polar fields form from the decay of tilted bipolar active regions. Flux transport dynamo models are self-excited in that they produce new activity cycles from old cycles by transporting decayed photospheric active region flux poleward and then transporting the same flux equatorward beneath the surface, amplifying it at active latitudes by differential rotation (in the Babcock-Leighton models), whereupon some of this flux emerges to create the new cycle. Surface flux transport models, by contrast, use sunspot number or magnetogram data as input for the new cycle. The photospheric fluid motions that break the active regions down and transport the flux poleward — diffusion, differential rotation and meridional flow — are essentially the same in both types of model. But flux transport dynamo models are generally (though not in all cases) axisymmetric, and they have depth-dependence in modeling both the latitudinal and radial components of the meridional flow as well as having a radially-varying turbulent diffusivity. Surface flux transport models more often include longitude-dependence, and describe the response of the radial field component to a latitude-dependent surface meridional flow and a constant surface diffusivity. We will discuss examples of both types of model in the following. But before discussing the kinematic models, we will begin with the initial phenomenological model of Babcock (1961).
3.1 Babcock’s phenomenological model for the solar cycle
Babcock (1961) developed a phenomenological model for the solar cycle that accounted for the reversal of the polar fields, the equatorward migration of sunspot emergence (Spörer’s law), Hale’s and Joy’s laws governing the polarities and tilts of sunspots (extended to bipolar active regions in general), and the fact that bipolar active regions appear to decay away. In this model, the interior differential rotation provides the energy that amplifies the interior field and produces the new activity cycle. While this model has been much modified by subsequent observational and modeling work, most of the basic picture has survived to this day.
Since, at the beginning of a cycle, the following-polarity flux in each hemisphere is preferentially of opposite polarity to the polar field in that hemisphere, and the trailing-polarity flux emerges poleward of the leading-polarity flux on average, this ongoing process weakens the polar fields and reverses them. The process continues until the end of the cycle, as the poleward expansion of following-polarity flux forms a new polar cap in each hemisphere, of opposite polarity to the one at the beginning of the cycle. The vast majority of the flux not involved in reversing and forming the polar fields is eliminated by the merging of expanded field from the active regions accompanied by reconnection in the atmosphere.
Babcock (1961) noted the lack of stability inherent in the model. The amplitude of a cycle is strongly dependent on the details of active region emergence — high-amplitude cycles would not necessarily be followed by high-amplitude cycles, and a strong polar field is not guaranteed to be reversed by a weak activity cycle, with consequences for the following cycle. But his model represented a major advance in our understanding of how the cyclical relationship between the active regions and the polar fields proceeds, and it also stimulated the first efforts to model the solar cycle kinematically.
3.2 Leighton’s numerical kinematic model for the solar cycle
The phenomenological solar cycle model of Babcock (1961) was based on an observed interrelationship between the active regions and the polar fields. His picture of the deformation and amplification of the Sun’s interior field by differential rotation, the emergence and decay of this field, and the cycle-dependent latitude separation of leading- and following-polarity flux that creates the polarity bias of the flux that expands to the poles, provided the first theoretical link between the activity and polar cycles, and still forms the basis of global dynamo modeling today. Two important gaps in the model were its lack of explanation for the latitudinal dispersal of surface active region flux (differential rotation can only expand the flux longitudinally), and a quantitative description of how the cycle may proceed. Both problems were addressed by Leighton (1964, 1969).
Leighton (1964) proposed a specific physical mechanism for the expansion and transport of decayed active region flux. Such a mechanism is essential for the preferentially following-polarity flux to reach the poles and change the field there, and also helped to explain the observed relative longevity of leading-polarity flux compared to following-polarity flux.
The supergranular convection flow field (Leighton et al., 1962) concentrates the decaying, large-scale field of an active region into narrow lanes at the boundaries of convective cells (Parker, 1963). Because these lanes are non-stationary (Leighton et al., 1962), they move an element of plasma around on the photosphere in a manner similar to a random walk. The solar plasma is highly conducting so that fields penetrating the photosphere must be dragged on this random walk. This process accounts for the relatively short lifetimes of individual flux elements in bipolar or unipolar flux patterns whose total fluxes are approximately conserved over much longer timescales. Treating magnetic concentrations as atom-like particles, Leighton (1964) assumed that adjacent positive and negative fluxes do not attract or repel each other but instead move independently of each other, dependent only on the fluid motions. Based on this random-walk process, Leighton (1964) developed a diffusion model for the breakdown and expansion of photospheric fields.
Leighton (1964) presented a formal solution of this diffusion model in spherical harmonics showing that diffusion time scales decrease with increasing multipole order. The smaller-scale structures tend to diffuse away, leaving the lowest-order structure, a dipole, which has a 5-yr decay timescale, approximately half a solar cycle. Leighton (1964) also presented numerical kinematic flux transport models for the decay by differential rotation and diffusion of active regions, featuring Joy’s law tilt and stronger leading- than following-polarity flux. The combination of differential rotation and diffusion resulted in poleward expansion in the form of two tilted, longitudinally stretched bands, one of each polarity, shown in Figure 21. The following-polarity band of flux expanded more and dominated the polar latitudes, consistent with the model of Babcock (1961). This diffusion model applied to multiple bipolar active regions leads to a dipolar field with sign reflecting the Joy’s law tilt bias of the bipoles. The model also reproduces the 90° phase lag between the polar-field and activity cycles first reported by Sheeley Jr (1964, see Section 4.4).
The sporadic character of sunspot formation and the resulting complex magnetic distributions indicate that there are significant irregularities in the subsurface fields. On the other hand, the long-term regularity of the solar cycle suggests that there is an underlying regularity in the field distribution that drives the cycle. Leighton (1969) focused on the regular component of the solar field by imposing constant or uniformly-varying models for the physical quantities within a relatively thin shear layer close to the solar surface. The resulting kinematic model reproduced the equatorward migration of active region flux emergence seen in magnetic butterfly diagrams, as well as the poleward migration of the zero contour of radial field, modeling the poleward march of the polar filaments that occurs before each polar field reversal (see Section 4.5).
Leighton (1964) imposed a critical field strength corresponding to that of the weakest sunspot fields then observed, a few hundred gauss. In his model, the emergence of azimuthal field decreases the strength of the remaining azimuthal field, an assumption consistent with the idea of Babcock (1961) that flux emergence would terminate at a given latitude when the interior toroidal flux rope became too fragmented. This idea was later challenged by Parker (1984), who argued that a net toroidal flux decrease would require a chain of active regions to erupt simultaneously over 360° of longitude. Whereas in Babcock’s model the equatorward migration of the flux emergence is a direct consequence of the differential rotation geometry alone, in Leighton’s model the emergence of flux at a given latitude strengthens the radial and meridional field components at adjacent lower latitudes and weakens them at higher latitudes, preferentially enhancing the toroidal field growth rate at lower latitudes. These assumptions would be challenged by subsequent modeling and observational work, but many of the critical components of Leighton’s model are still in use.
3.3 The influence of meridional flows on the Babcock—Leighton model
Kinematic flux-transport modeling took major steps forward in the 1980s, stimulated by new observational evidence of meridional circulation flows in Doppler measurements by Duvall Jr (1979) and Labonte and Howard (1982), along with evidence of subsurface equatorward return flow in torsional oscillation patterns found by Howard and Labonte (1980). These observations were followed by photospheric feature-tracking (Komm et al., 1993), Doppler (Ulrich, 1993) and helioseismic observations (Hathaway et al., 1996) of poleward surface meridional flows of 20 ms−1. The observations provided critical guidance for modelers of the solar cycle.
Sheeley Jr et al. (1985) extended the photospheric flux transport work of Leighton (1969) by adding a background meridional flow explicitly to the model, and by modeling the flux transport in the two surface spatial dimensions over an entire solar cycle and comparing the results with observations of photospheric and interplanetary mean field strengths and coronal hole distributions. They found reasonable agreement between the model results and the observations. The main effect of the meridional flow was to weaken the mean photospheric field by disrupting long-lived patterns of strong photospheric flux, and by removing active region flux from active to polar latitudes.
Wang et al. (1989) presented an azimuthally and radially averaged kinematic flux transport model similar to the one developed by Leighton (1969), but with the Sheeley Jr et al. (1985) meridional flow model included. They applied this model to study the combined effects of super-granular diffusion and meridional flow on the development of polar fields. They found that the meridional flow plays an important role in carrying the trailing-polarity flux poleward, and concentrating the polar fields at the poles against the spreading effect of diffusion. Whereas diffusion alone would tend to produce a dipole-like distribution (see Leighton, 1964, and Section 3.2), the observed distribution is much more concentrated (see Section 2.2), of the form cosnθ, where n is in the range 8–11, and polar fields with this degree of spatial concentration can only be achieved by including significant poleward surface meridional flow. They also found that, to reproduce the very intense poleward surges of decayed active region flux and major polar field changes observed during 1980–1982, they needed both accelerated meridional flow and enhanced bipole emergence rates.
It is not known whether the meridional flows really accelerated to higher speeds during 1980–1982 to produce the large polar field changes observed during this time. The fact that Wang et al. (1989) had to multiply the observed flux emergence rate by 1.6 to reproduce the observed polar reversal suggests that other factors were involved in producing these surges, such as changes in the bipoles’ size or Joy’s law tilt distribution. In the model of Wang et al. (1989) the latitude separation between the positive and negative polarities in each hemisphere was kept constant, but this parameter may vary over time and might produce large effects on the poleward surges of flux and the polar fields themselves.
Wang and Sheeley Jr (1991) studied the emergence and subsequent evolution of bipoles in some detail. They suggested that Coriolis forces acting on a rising, expanding flux loop can account for the observed magnitude and latitudinal variation of bipole tilt angles. They then investigated the effects of the large-scale photospheric flows on the emerged bipoles. Differential rotation has no effect on the axisymmetric component of the field, including the axial dipole moment. The discrepancy between the dipole moments that the bipoles emerge with and the subsequent dipole moment of the photospheric field is therefore due to diffusion and meridional flow.
Leighton (1964) — see Section 3.2 — showed that in the presence of diffusion alone the magnetic multipoles would decay such that the low-order multipoles decay slowest and the lowest-order multipole, the dipole, would decay on a timescale comparable to a solar cycle. On the other hand, as Wang and Sheeley Jr (1991) agued, for a poleward meridional flow in the absence of diffusion, a bipole not straddling the equator would be transported to the nearest pole, and its two polarities would merge there or in transit. Thus, either diffusion or meridional flow acting alone would reduce the dipole moment of an emerged bipole.
Because of this effect, diffusion and meridional flow can cause the axial dipole moment of a bipole to increase or decrease after emergence, depending on the bipole’s latitude, but in general the effect of the flows is to decrease a bipole’s dipole moment. Over most of the active latitude range and with solar-like diffusion rates and meridional flow speeds (10–20 m s−1), the diffusion and meridional flow can be relied upon to reduce the dipole moment of a bipole. The conversion from toroidal to poloidal flux in the Babcock-Leighton model is therefore due to the tilt angles with which the bipoles emerge and not to the subsequent flux transport. The flux transport only serves to redistribute the poloidal bipole flux component poleward and concentrate the polar fields.
Wang et al. (1991) presented a development of the flux transport model including not only surface flux transport by super granular diffusion and poleward meridional flow but also the effects of subsurface turbulent diffusion and equatorward return flow. As in Babcock’s and Leighton’s models, differential rotation shears subsurface poloidal fields to produce strong toroidal fields. Unlike Babcock and Leighton, Wang et al. (1991) did not assume that toroidal flux emergence decreases the toroidal flux beneath the surface. Instead, the subsurface toroidal field is destroyed by unwinding the toroidal field during the declining phase of each cycle due to the poloidal field having switched polarity, and by subsurface diffusion as the toroidal flux is transported equatorward by the subsurface meridional flow and merged with oppositely-directed toroidal flux in the opposite hemisphere.
Meridional circulation also allows the equatorward migration of flux emergence to occur without requiring a radial gradient in the interior differential rotation profile. Helioseismic results from Thompson et al. (1996) and Schou et al. (1998) showed that radial gradients in subsurface angular velocity are small in the convection zone down to around 0.7 solar radii, and that the large radial shears that occur around 0.7 solar radii are such that the rotation rate decreases with depth at low latitudes, contrary to the assumption of Leighton (1969). Unlike the model of Leighton (1969), the Wang et al. (1991) model behaved in a stable, oscillatory manner without a radial gradient in the interior angular velocity.
In this model, as in the radially and azimuthally averaged model of Wang et al. (1989), the polar fields underwent less sudden reversals than the solar polar fields are observed to do. The authors attributed this difference to the fact that the meridional flow speed in the model was independent of time, whereas the solar meridional flow speeds may vary significantly. This topic has become particularly important recently in the context of efforts to explain the weak polar fields of cycle 23 — see Section 3.6.
3.4 The evolving, “synchronic” synoptic map
We now discuss a very useful application of photospheric flux transport modeling: the creation of time-dependent, full-surface synoptic photospheric magnetograms. Since the 1960s and 1970s (Howard, 1967; Schatten et al., 1969; Harvey et al., 1980), synoptic maps for the photospheric magnetic field have been constructed using a fixed coordinate system defined by Carrington, where the vertical axis represents heliographic latitude or sine(latitude) and the horizontal axis represents both longitude and time, a full rotation corresponding to both 360° and 27.27 days, based on the rotation rate of latitudes ±26°, representative typical latitudes of sunspots, as observed from Earth. Each full-disk magnetogram is remapped from sky-image coordinates to heliographic coordinates. The remapped images are then positioned in the Carrington coordinate grid and either pasted together or averaged together in some fashion, often giving more weight in the averaging to measurements near central meridian than to measurements near the east and west limbs. In these maps the magnetic field at any given location on the photosphere remains unchanged as long as that location is not observed. Because of this, different longitudes are represented by observations taken at different times, up to 27.27 days apart. In this way such a map is “diachronic”. This is how the synoptic maps discussed in Section 2 were created. Maps of this kind have proved to be very useful in the past. The slowness of the dispersal and transport of active region flux, and the slow evolution of the polar fields allow even such infrequently updated maps to represent the full-surface field reasonably well most of the time. On the other hand, the development of a time-dependent, “synchronic” map, a map of the full surface as if it were observed all at a single time, is clearly a necessary step towards more accurate synoptic maps and extrapolated models for atmospheric magnetic fields.
As the basic concepts of photospheric flux transport have become more established, increasingly realistic models for the photospheric magnetic field distribution have been developed in the form of synoptic, full-surface maps of the photospheric field. Unlike the “diachronic” synoptic maps described above and featuring elsewhere in this review, (e.g., in Sections 2.6 and 4.5), these “synchronic” maps are designed to provide a snapshot of the full-surface photospheric field distribution at any time. Since these maps necessarily include longitudes that have not been observed for two weeks, and polar latitudes that remain unobserved for months at a time, a model for the photospheric flux transport seems to be a necessary condition for an accurate full-surface snapshot.
Worden and Harvey (2000) developed a model for two-dimensional photospheric flux transport, with physical ingredients similar to those of Leighton (1969) and Sheeley Jr et al. (1985), and that could be used to derive an instantaneous snapshot of the global photospheric field at any given moment. In this model, however, new observations are added to the portion of the map representing the front side of the Sun, while unobserved fields are evolved according to the flux-transport model.
In the transformation of the full-disk images from sky-image coordinates to heliographic coordinates, the number of sky image pixels per heliographic remap pixel is not constant. This ratio, an information density, is maximum at disk-center and decreases towards the limb. This information density distribution is taken into account when new observations are added to the map. When different observations of a single solar location are averaged together, the average is weighted by the information density, so that data observed close to central meridian contribute most to the map. In most implementations a further weighting is applied favoring central meridian data to account for the superior sensitivity of the measurements near disk-center. In this calculation, the polar latitudes have relativity low information density and sensitivity, but central-meridian polar data still have highest weighting at these latitudes.
Worden and Harvey (2000) did not include sky image pixels beyond a certain distance from disk-center. This means that observations for the field at each pole are only included in the model over about four months per year. The polar regions are filled with fields transported poleward by the flux-transport model over the remainder of the year. The modeled polar fields were found to be most sensitive to the meridional flow speed, consistent with the flux transport modeling discussed in Section 3.3.
The model includes the Snodgrass (1983) differential rotation law and a meridional flow profile similar to that of Sheeley Jr et al. (1985). It departs from the standard flux-transport model of Sheeley Jr et al. (1985) in two important ways. A random Gaussian distribution of magnetic flux with mean field strength 1.8 G is continuously added to the model to sustain the quiet-Sun magnetic fields, and so that their dispersal can be reproduced throughout the photosphere. Without this ingredient the photospheric network field would disappear through flux cancellation in 2–3 days (Schrijver et al., 1997).
Also random attractors are used to model the dispersal of the fields instead of the diffusion model based on random walks introduced by Leighton (1964). Supergranular convective cells are observed to be randomly distributed across the solar surface. The cell flows clearly transport magnetic flux not in a diffusion-like manner in all directions but preferentially towards the cell boundaries and their junctions. Worden and Harvey (2000) therefore modeled these processes using random attractors. Unlike the random-walk diffusion model of Leighton and Wang and Sheeley, a random attractor model clumps magnetic flux together in many small concentrations as seen in observations. Worden and Harvey (2000) created an attractor matrix of (360 × 180 pixels, the same size as the map itself, and assigned each attractor a random number such that the mean attractor number per unit solar area was conserved. For each pixel in the map a search space was defined as all neighboring pixels within a supergranular scale (≈ 13 500 km). The distance, direction and strength of the largest attractor in the search space determined the displacement vector of the pixel’s flux due to the convection. This vector was combined with those associated with the differential rotation and meridional flow to compute the displacement of the pixel’s flux over each time step.
Schrijver (1989) showed that active-region flux is more resistant to dispersal than the weaker surrounding flux. This apparent magnetoconvective coupling between the fields and flows stimulated a new effort to simulate photospheric flux evolution and polar field development with greater realism (Schrijver and Title, 2001, see Section 3.5). Worden and Harvey (2000) applied simulated supergranular motion only to flux weaker than 25 G. The optimal effective diffusion coefficient in their model was 520 km s−1.
Adding the small-scale background flux maintains the small-scale features of the field at low latitudes against diffusion, but does not alter the dispersal of large-scale features (see also Wang and Sheeley Jr, 1991). Compared to Figure 24 (bottom), the active regions in Figure 25 have dispersed more quickly because of the flux dispersal model, and the flux has a network-like appearance because of the random-attractor form of the flux dispersal. The modeled polar fields are also much more textured than those in the Kitt Peak synoptic maps in Figure 24, more reminiscent of the high-resolution Hinode observations shown in Section 2.1. The polar fields of an evolved synoptic map are unipolar, unlike the Hinode observations described in Section 2.1. The small-scale polar features in Figure 25 can be identified with the large, cycle-dependent class of patches discussed in Section 2.1 that are generally of the same polarity and define the polarity of the polar cap.
The addition of new observations significantly improves the resemblance of the model to the observed field for CR 1929 (Figure 24, middle). The flux transport alone cannot reproduce the disappearance of the small active region in box 3, or the emergence of new active regions. But updates from daily observations successfully catches the disappearance of flux in box 3, and some of the flux emergence elsewhere. Clearly, the model cannot be updated at all longitudes at all times, but the regular additions of new magnetograms keep the model as up-to-date as possible.
Synoptic map construction, whether of traditional Carrington maps or of maps derived from flux-transport models described above, usually assimilate new magnetograms by direct insertion into the model in a weighted average or by blending observations with model data. Such methods do not quantitatively take into account the relative accuracy of the new observation compared to the observations already present or the model.
Using the Los Alamos National Laboratory data assimilation framework, Arge et al. (2011) have developed a version of the model of Worden and Harvey (2000) model, named Air Force Data Assimilative Photospheric Flux Transport (ADAPT), that includes statistical filtering of new data. The filtering methods include ensemble least squares and Kalman filtering. Arge et al. (2011) ran their model over half a solar cycle (2003–2009) using ensemble least squares filtering, and found encouraging agreement between the modeled polar field strengths and MWO polar field measurements taken with optimal B0 angle. They have also developed forecasts of the solar radio flux at 10.7 cm, known as the F10.7 flux, using absolute field strengths from their flux-transport modeling, and improved reconstructions of coronal hole boundaries from potential-field source-surface models. In a case study, Arge et al. (2013) included helioseismic data for active region emergence and development on the far side of the Sun and they reported further improvement in the performance of their model in this case.
3.5 Modifications to random walks due to magnetoconvective coupling
Additional observed properties of the photospheric field were incorporated into the flux transport model of Schrijver (2001). In this work the flux was modeled by an ensemble of concentrations represented by discrete point sources undergoing collisions and fragmentation. Large active regions and small ephemeral regions were included. A flux-dependent field dispersal was applied to recreate the initial decay of active regions, including a reduced rate of this decay seen in observations for strong magnetic fields, believed to be caused by magnetoconvective coupling (Schrijver, 1989).
In mature active regions, the average observed flux density remains around 100–150 G, independent of the region’s age or size (Schrijver and Harvey, 1994). This property is clearly inconsistent with the standard linear random-walk diffusion model of Leighton (1969) and subsequent authors (Sections 3.2–3.3). Whereas the random walk model is remarkably successful at redistributing the large-scale photospheric field in agreement with observations, the necessary diffusion rate of 600 ± 200 km2 s−1 (e.g., Wang et al., 1989; Durrant and Wilson, 2003) is significantly higher than the rate indicated by flux concentration tracking observations.
In his flux transport model, Schrijver (2001) combined the models for collision, fragmentation and magnetoconvective coupling with differential rotation and meridional flow profiles empirically derived by Komm et al. (1993) from NSO/KP magnetograms. Schrijver and Title (2001) proceeded to use this model to study the evolution of polar fields of the Sun during solar cycle 21, and also in non-solar stars including the case of a star 30 times more active than the Sun.
3.6 Unusual cycle 23 minimum
Since the beginning of the space age over 40 years ago, when detailed observations of the solar magnetic field began, only four full solar cycles have occurred, cycles 20–23, and a fifth, cycle 24, is in progress. During the minimum of cycle 23 it became apparent that the Sun was behaving like it had not been observed to behave previously. While the photospheric flux transport between the active and polar latitudes is complicated by the interaction of multiple processes, the strength of the polar fields that develop during a cycle is generally expected to be related to the amplitude of the activity during the cycle, measured by, e.g., the sunspot number. According to the sunspot number or the equatorial dipole or other non-axisymmetric multipole components, cycle 23 was about 30% weaker than the two previous cycles (the smoothed maxima were 164.5, 158.5, 120.8 for cycles 21–23, respectively, making cycle 23 27% weaker than cycle 21 and 24% weaker than cycle 22) but the polar fields were about 40% weaker than they had been during the previous 3 minima (Svalgaard and Cliver, 2007). The cycle 23 minimum was also unusually quiet: there were 265 and 261 spotless days in 2008 and 2009, respectively, beaten only by 1878, 1901 and 1913 since 1849 (Jiang et al., 2013). We will discuss some consequences of this unusual behavior in Section 4.
Various explanations for the weakness of the cycle 23 polar fields have been suggested. The polar field strength at the end of a given sunspot cycle depends on several things: the polar field strength at the beginning of the cycle, the total amount of active region flux that emerges during the cycle, the axial tilts and latitudes of the active regions, the supergranular diffusion rate, and the meridional flow speed profile in time and latitude. The two surface flux-transport parameters with arguably the greatest influence over the polar fields are the Joy’s law tilt and the meridional flow speed, which between them mostly determine the polarity bias of the decayed active region flux sent poleward.
Dikpati (2011) used simple numerical estimates and detailed kinematic dynamo modeling to show that even a quite modest decrease in active region field from one cycle to the next, such as between cycles 22 and 23, could produce a large decrease of polar field strength. The amplitude of the polar fields is very sensitive to the details of the strength and decay of the active regions. To see this, suppose that at the end of a given cycle the polar fields have strength one unit. Then it would take a change of minus two units to reverse the polar fields to the same strength. But if the active regions contribute just 20% less flux than is necessary to do this in each hemisphere, then they will still reverse the polar fields but with a 40% drop in the polar field strength compared to the initial strength, a decrease twice as large as the decrease in decayed active region flux (Dikpati, 2011). Indeed, generally speaking, the presence of active region fields is correlated with changes in the polar fields as can be seen by comparing Figures 17 and 19. However, there are intervals of time when significant quantities of active region field are present in the photosphere but the polar fields do not change significantly. In particular, between 2002 and 2006 there were significant active region fields in the photosphere while the polar fields remained remarkably constant. Because there are periods of time when there are active region fields on the Sun that produce no detectable effect on the polar fields, Dikpati’s analysis does not fully explain the weakness of the polar fields.
In Babcock-Leighton flux transport, there are two possible explanations for unchanging polar fields in the presence of active regions. According to both explanations, decaying active fields still reach polar latitudes during these time intervals but these decayed active region fields are of such mixed polarity that their net effect on the polar fields is approximately zero. Figure 17 shows that the plumes of decayed active-region flux moving poleward have been of more mixed polarity since the cycle 23 polar reversal than before. One explanation is that the meridional flows are so fast that the leading polarities in the two hemispheres do not have time to interconnect and interact with each other before being swept poleward (e.g., Schrijver and Liu, 2008; Wang et al., 2009; Nandy et al., 2011; Jiang et al., 2013). An alternative explanation is that the active region Joy’s law tilts changed their hemispheric bias during cycle 23 (e.g., Jiang et al., 2013; Petrie, 2012). In this scenario approximately equal quantities of each polarity would be sent poleward with approximately zero net effect on the polar fields even for slow meridional flow speeds.
Several important ingredients of flux transport models are not well constrained by observations. Unfortunately, these include the meridional flow and the Joy’s law tilt. The meridional flow results often depend on the method used to derive them, e.g., Doppler measurements, magnetic feature tracking and helioseismic inversions of different kinds (global modes, ring diagrams). Surface Doppler (Ulrich, 2010) and helioseismic (Basu and Antia, 2010) meridional flow measurements generally agree with each other, having peak speeds at low latitudes around 25° on average, whereas magnetic feature tracking meridional flow measurements (Hathaway and Rightmire, 2010) have much lower values peaking at higher latitudes, around 50°. Magnetic feature tracking methods give a peak at higher latitudes because they do not separate the bulk fluid flow from the effect of supergranular diffusion. Poleward of active latitudes the diffusion generally acts in the same direction (i.e., poleward) as the meridional flow (Wang et al., 2009; Dikpati et al., 2010). Also, the Joy’s law tilts of sunspot pairs and bipolar active regions are difficult to measure in a comprehensive and objective way, and the results vary (e.g., see the discussion in McClintock and Norton, 2013). Definitions of sunspot pairs often exclude important strong field structures that can contribute to the Babcock-Leighton mechanism, and the details of the tilt calculation vary from study to study.
Meridional flow perturbations local to the activity bands may also play a role in modifying the flux transport. Sun et al. (2015) found an anti-correlation between active region field strength and the mid-latitude poleward flow speed of the associated decayed field, consistent with a field-dependent converging flow towards active regions (Zhao and Kosovichev, 2004). Using a flux-transport model, Jiang et al. (2010) studied the effect of meridional flow perturbations caused by near-surface inflows towards the active region band in each hemisphere, and found that large perturbations reduce the tilt angles of bipoles, thus reducing their contribution to the polar field changes. The amplitude of the meridional flow perturbations are larger for stronger solar cycles, consistent with the anti-correlation between Joy’s law tilt and cycle amplitude reported by Dasi-Espuig et al. (2010).1 Such inflows can strongly affect the behavior of flux-transport models (Cameron and Schüssler, 2012).
Jiang et al. (2013) performed a study that investigated the free parameters of the flux transport model. In this model the bipolar active region emergence was proportional to the monthly sunspot number, with the position, area and tilt angle of each bipole prescribed by an empirical model with random scatter. A model for preferred longitudes, as exhibited by activity complexes (see Section 4.5), was also included. They also imposed cycle-average bipole tilt angles that were inversely proportional to cycle amplitudes, consistent with the results of Dasi-Espuig et al. (2010). They applied the differential rotation profile from Snodgrass (1983) and a meridional flow profile from van Ballegooijen et al. (1998) designed to vanish poleward of ±75° (to preserve filament channels — see Section 4.5). They compared their modeled polar fields to measurements from WSO, and the model agreed well for the cycle 21 and 22 polar fields but was a factor of about 2 too strong compared to the measured polar fields of cycle 23.
Thereupon Jiang et al. (2013) explored the possible causes of the discrepancy by examining the effects of varying each free parameter in the model. Since the diffusivity models the effect of supergranular motion and is not expected to change significantly from cycle to cycle, Jiang et al. (2013) focused on changing the bipole emergence rate, the meridional flow and the bipole tilt. They found that a 40% reduction in the emergence rate, a 28% decrease in the average tilt angle and a 55% increase in the meridional flow speed for cycle 23 resulted in agreement between the modeled and observed polar field strengths for cycle 23. However, the reduction in the emergence rate also resulted in a major reduction in the open flux for cycle 23 (derived from PFSS modeling), and the reduction in the mean bipole tilt produced a 1.5-yr delay in the polar field reversal, whereas the increased meridional flow produced better agreement with the observed open flux and polar reversal time. Therefore, three independent studies, Schrijver and Liu (2008), Wang et al. (2009) and Jiang et al. (2013) all point to the same conclusion: that the likeliest cause of the weak polar fields of cycle 23 was an increase in the meridional flow speed for cycle 23.
The fact that Jiang et al. (2013) required a very large 55% increase in the meridional flow speed, compared to the much more modest 10–15% reguired by Wang et al. (2009), is due to the much smaller low-latitude gradient in the flow profile used by Jiang et al. (2013). Schrijver and Liu (2008) were able to control their polar fields by varying the low-latitude flow gradient alone. Indeed, the very large gradient of the profile used by Wang et al. (2009) may have played an exaggerated role in their results.
Indeed, not only the low-latitude gradients but the overall shapes of the meridional flow profiles are also crucial in determining the duration and timing of solar cycles, including the reversal of the polar fields. Dikpati (2011) studied how the location of the peak flow speed influences the effect of meridional flow speed changes on the polar fields. She argued that when the meridional flow speed peaks at low latitudes, an increase in the overall speed produces a faster poleward transport of the leading polarity compared to the trailing polarity, enhancing the cancellation between them, resulting in weaker polar fields. This is matches the surface flux-transport results of Schrijver and Liu (2008), Wang et al. (2009) and Jiang et al. (2013). If, on the other hand, the flow speed peaks at higher latitudes, then an increase in the flow speed causes the trailing polarity to speed up relative to the leading polarity, inhibiting cancellation. This can result in stronger polar fields if enough trans-equatorial diffusion is allowed to occur (see Section 3.3).
To affect the polar fields significantly, the meridional flows changes must occur at or equator-ward of active latitudes, as Dikpati (2011) emphasized. The meridional flow speed measurements of Ulrich (2010), Basu and Antia (2010) and Hathaway and Rightmire (2011) do not show evidence of significantly faster flows at active latitudes during cycle 23 than during previous cycles. Moreover, Basu and Antia (2010) concluded that the flow speed variation reported by Hathaway and Rightmire (2010) was associated with a flow pattern that migrated equatorward with the magnetic activity belts. When this pattern was removed from their data, the speed variations vanished. This evidence argues against the flow-related explanations offered by Schrijver and Liu (2008), Wang et al. (2009) and Jiang et al. (2013). The weak polar fields may instead have something to do with a change in the flux emergence patterns, in particular the patterns of bipole tilt angles.
Direct attempts to find changes in average bipole tilt angles have not produced an explanation for the weak polar fields. Schrijver and Liu (2008) selected bipolar active regions observed between 1997 and 2008 by MDI for at least 7 days on the solar disk, and that were at least 30° away from any other region throughout this time. They arrived at a set of 136 regions that met these criteria. They calculated the centers of gravity of the positive and negative fluxes of each region and thereby estimated the Joy’s law tilt angles of the regions, taking geometrical and projection effects into account. They found no systematic change in the tilt angle over a period of up to 8 days on the disk. They did not report a change of mean tilt angle over the cycle. Stenflo and Kosovichev (2012) have analyzed the Joy’s law tilt angle of selected magnetic bipoles over time and found no statistical change in average Joy’s law tilts, and Li and Ulrich (2012) found from a long-term study that tilt angles of spots appear largely invariant with respect to time at a given latitude, but they decrease slowly during each cycle following the butterfly diagram (see McClintock and Norton, 2013, on the difficulty and complexity of sunspot tilt measurements).
Recall that in the flux transport simulations of Jiang et al. (2013), the weak cycle 23 polar fields could also be reproduced by a 28% decrease in the average tilt angle of sunspots, but this would also lead to a 1.5-year delay of the cycle 23 polar field reversal that was not observed. As we will see in Section 4.4, Muñoz-Jaramillo et al. (2013) also stressed the importance of accounting for varying tilt angles in such calculations. Petrie (2012) found that the latitude centroids of the positive and negative active region fields converged in each hemisphere around 2003, implying a much-diminished active-region poloidal field around this time — see Figure 18. In this calculation, the active regions’ contributions to the polar field are found to have a good proxy in the product of the latitude displacement of the positive and negative active region flux centroids and the total active region flux. This quantity was found to track the high-latitude poleward flux surges, which in turn tracked the polar field changes. Thus, the convergence of the positive and negative active region flux centroids in each hemisphere around 2003 could be linked to the lack of major polar field change between this time and the beginning of cycle 24. At the same time as the positive and negative active region flux centroids began to converge in each hemisphere, the high-latitude poleward surges of field were observed to lose their polarity bias in each hemisphere and the polar fields stopped strengthening, as seen in Figure 19. These two related patterns are consistent with the Babcock-Leighton model. Recently, Jiang et al. (2015) found a similar result from surface flux-transport simulations. These results suggest that changes in the meridional flow speed or shape, though potentially influential, may not be necessary to explain the weakness of the polar fields during cycle 23, and that a change in the latitudinal distribution of the active region flux may have been responsible.
Another notable development during cycle 23 was the increased asymmetry between the north and south hemispheres. While time-dependent asymmetry between the two hemispheres is generally not uncommon, the asymmetry that developed during cycle 23 was stronger than previously found since routine magnetogram observations began in the 1960s and 1970s. Figure 17 shows that, whereas the two hemispheres had approximately equal levels of activity throughout cycles 21 and 22, during the decline of cycle 23 the southern hemisphere became much more active than the northern hemisphere, and the north became more active than the south when cycle 24 began, since when the south has again become more active than the north. The physical relationship between the two hemispheres remains mysterious, but idealized models for the solar dynamo can be used to explore this question.
4 Polar Fields and the Solar Atmosphere
4.1 The effects of the polar fields on global coronal magnetic structure
The Sun’s polar fields have long been known to have a profound influence on the global structure of the solar atmosphere. There is a close relationship between the polar fields and the axial dipole component such that the polar fields dominate the global coronal structure over much of the cycle. Hoeksema (1984) used WSO synoptic magnetograms and PFSS models to demonstrate that, during most of the solar cycle, the quadrupole and occasionally octupole moments of the field are important for the large-scale structure of the coronal field, including the structure in the ecliptic plane, producing tilts and warps in the streamer belt and equatorial current sheet and creating low-latitude coronal holes. He concluded from a study of the total strengths of the various multipolar orders that the complex field evolution near maximum does not correspond to a dipole rotating from north to south as the polar fields reverse polarity, as had been previously suggested. The rotating dipole interpretation does not fit well with the picture of a rather complex and inefficient process presented in previous sections, of the polar fields being built gradually from the accumulation of small parcels of decayed active region flux with a polarity bias, before being nibbled away again by further parcels of decayed active region flux with opposite polarity bias.
Figure 33 shows three “hairy-ball” plots of the coronal magnetic field from the years 1996, 2000 and 2009. The plots therefore represent the cycle 22 minimum, and cycle 23 maximum and minimum, respectively. The differences between the minimum plots and the maximum plot clearly emphasize the much simpler, nearly axisymmetric axial dipole structure associated with dominant polar fields during solar minimum, and the complex three-dimensional structure that is produced by multiple active regions in the absence of strong polar fields.
During the long cycle 23/24 minimum the axisymmetric dipole and octupole components were about 40% weaker than during the previous cycle minimum, whereas the non-axisymmetric multipoles did not become so comparatively weak until 2009. This also resulted in a more complex coronal structure with more low-latitude coronal holes during the cycle 23/24 minimum than during the previous minimum (de Toma, 2011). We will discuss this topic further in Section 4.2.
4.2 Coronal holes
That low-density regions are present in the solar atmosphere, and are particularly prominent at the Sun’s poles over most of the cycle, has been known for many decades (Waldmeier, 1957), and it has become increasingly clear that these “coronal holes” are associated with open magnetic flux (see the review by Harvey, 2013), represented by the green and red lines in Figure 33. Polar coronal holes have been studied using limb observations of the Fe xiv 5303 A coronal green line (Waldmeier, 1957), K-coronagraph observations (Bravo and Stewart, 1994), X-ray and EUV observations (Broussard et al., 1978), He i 10830 Å spectroheliograms (Harvey et al., 1975) and PFSS (Schatten et al., 1969; Altschuler and Newkirk, 1969), magnetohydrodynamic (e.g., Linker et al., 2011) and magnetofrictional (Mackay and Yeates, 2012) models.
Coronal holes appear in several forms and for a variety of different reasons. Using NSO Kitt Peak He i 10830 A spectroheliograms and longitudinal photospheric magnetograms, Harvey and Recely (2002) identified three classes of coronal hole: polar coronal holes confined to high latitudes (> 60° and < −60°), isolated coronal holes at active latitudes associated with the remnants of decaying active regions, and transient coronal holes that briefly form after coronal mass ejections. The accumulation of nearly unipolar flux from the decay and poleward transport of tilted active regions is clearly the cause of the polar coronal holes. The competition between meridional flow and diffusion at high latitudes strongly influences the extent of the polar coronal holes. In the absence of meridional flow the polar field would have a dipolar distribution, and the polar coronal holes would extend to around ±40° instead of around ±60° for the much more concentrated cosnθ with n = 8–11 as observed (Sheeley Jr et al., 1989).
Harvey and Sheeley Jr (1979) found that, early in the cycle, the decaying leading polarities of active regions can create locally unbalanced flux patterns, which tend to form coronal holes ahead of active regions, while the following flux travels poleward to cancel with the polar field. During the declining phase it is the following flux that tends to open. Petrie and Haislmaier (2013) showed that coronal holes generally open without changing the global coronal topology, at least in the presence of significant polar fields: in the 14 examples that they studied, the hole always formed with polarity matching the polar hole on the side of the streamer belt where the region decayed, demonstrating the dominance of the polar fields over the coronal structure.
Over most of the solar cycle the largest coronal holes are located at the poles, as in the top left and bottom plots in Figure 33. While not as active as and much weaker than the active region fields, the polar fields have great influence in the heliosphere via the polar coronal holes. The large spatial scale and unipolarity of the polar fields prevent these fields from connecting back to the surface before they reach great heights, where the field is too weak to close against the pressure of the expanding solar wind. Because the solar wind is accelerated to super-Alfvénic speeds low in the atmosphere, and the coronal plasma is highly conducting, at a height of around a solar radius the field becomes too weak to pull the magnetic field lines closed, and the solar wind drags the field outward and opens it to the heliosphere (Parker, 1958). During times of maximum activity, and particularly when the polar fields are reversing, the coronal holes are approximately equally spread over all latitudes, with no dominant polar holes, as in the top right plot in Figure 33.
4.3 Relationship between coronal hole structure and solar wind speed
It was discovered during the 1970s from Skylab data that high-speed solar wind streams originate from large coronal holes (Krieger et al., 1973; Nolte et al., 1976; Zirker, 1977). Wang and Sheeley Jr (1990) investigated the empirical relationship between solar wind speed patterns and the magnetic field structure at the solar wind source location. Using solar wind data from a series of spacecraft, they showed that the bulk solar wind speeds tend to be significantly lower during solar maxima than during solar minima. Polar passes of the Ulysses spacecraft have since shown the fundamental differences between the solar wind distributions during solar minimum and maximum conditions.
Wang and Sheeley Jr (1990) showed that the unsigned interplanetary field and the solar wind speed are not highly correlated but that the fraction of solar surface area covered by open magnetic flux, estimated using PFSS models, is better correlated with the wind speed. As the low-latitude coronal hole areas become small and the polar holes become dominant during solar minimum, the solar wind speeds become high. Crucially, a strong correlation was found between the average unsigned photospheric field strength in open-field regions and the solar wind speed at 1 AU.
The results suggested that the high-speed streams are indeed associated with low areal expansion rates in the corona. The high-speed streams originated mainly from the boundaries of the polar coronal holes, often from extensions that form when decayed active region flux of the same sign as the local polar field is transported poleward, particularly during the declining phase of the cycle. The fast streams can also come from small, detached coronal holes. If a small hole is close to a large hole of like polarity then the small hole can be prevented from expanding by the volume-filling nature of the large hole, in which case the small hole can produce a high-speed stream. In contrast, solar maximum brings a lull in the solar wind speed. During solar maximum the wind comes from small, low-latitude coronal holes. Because the total photospheric area of open flux is small, these small coronal holes generally expand rapidly between the photosphere and the source surface, resulting in low solar wind speeds.
Arge and Pizzo (2000) extended this model by deriving a continuous empirical function relating two parameters, the magnetic expansion factor, fs, and the angular separation between the open field footpoint and the boundary of the nearest coronal hole, to the solar wind speed at the source surface. They also added a simple kinetic model for solar wind propagation between the source surface and Earth, and improved the application of boundary data to the model. This model, now called the Wang-Sheeley-Arge model, is still used in routine solar wind prediction.
The second orbit caught the rise and maximum of cycle 23 and showed a complex solar wind structure at all latitudes, including streamers, coronal mass ejections, and small coronal holes. Covering the decline and minimum of cycle 23, the end of Ulysses’ second orbit and the third orbit found the heliosphere in a simpler state but with significantly more complex structure than observed during the first orbit. There was a more complicated current sheet structure with both greater tilt with respect to the equator and a less planar belt of low-speed flow. This result is consistent with the heliospheric current sheet tilt patterns in Figure 35, where the cycle 22 minimum current sheet structure was flatter than the cycle 23 minimum structure. The tilt of the heliospheric current sheet was significantly higher during the cycle 23 minimum than the cycle 22 minimum, at east until the sunspot number became unusually low in 2009 after the end of the Ulysses mission. The band of solar wind variability detected by Ulysses also extended to higher latitudes during cycle 23 minimum than during cycle 22 minimum. The simple bimodal solar wind structure of the left and right plots in Figure 40 corresponds to the simple dipolar structure in the top left and bottom plots in Figure 33, and the more complex middle plot of Figure 40 to the top right plot of Figure 33.
The solar wind parameters also showed different behavior during the third orbit compared to the first (McComas et al., 2008). The fast solar wind was about 3% slower, about 17% less dense, about 14% cooler and had about 20% less mass flux. The dynamical and thermal pressures were also significantly smaller (by 22% and 25%, respectively), prompting the authors to speculate that the heliosphere may have shrunk between the two orbits.
Measurements of the solar wind parameters in the ecliptic plane from the ACE spacecraft over the same period quantitatively matched the Ulysses observations and showed identical trends, indicating significant long-term variation at all latitudes. The implied reduction of energy and mass flux below the solar wind sonic point is consistent with a reduction of the open solar magnetic flux over this time interval.
Sections 4–4.3 have shown that the global structure of the solar atmosphere has changed between the cycle 22 and 23 minima, corresponding to decreases in the polar field strength and changes in activity patterns discussed in Sections 2–3. We will next discuss the properties of smaller magnetic features whose behavior is intimately related to the polar fields.
4.4 Polar faculae
The previous subsections discussed the global influence of the polar fields over the solar corona and heliosphere. These observations and models emphasize the importance of the polar fields and how urgently we need to understand them. Alternative ways to study the polar fields are based on smaller-scale solar structures whose connections to the polar fields help us to understand these fields’ behavior and influence. We will discuss such phenomena in the next few subsections.
In Section 4.2, we discussed the correlation between microwave brightness temperature in the polar coronal holes and the polar magnetic field strength. This correlation allows us to infer connections between the surface polar flux and conditions in the atmosphere, and it also allows us to validate or supplement a difficult magnetic field observation using an independent data source. Other sources of information on the polar fields come from smaller-scale structures in the atmosphere that trace the progress of the polar fields, in particular polar faculae, seen in white-light photospheric images, and chromospheric filaments seen in, e.g., Hα or He i 10830 Å images. Observations of faculae and filaments also extend significantly further back in time than systematic magnetograph measurements. We will discuss faculae in this section and filaments in the following section.
4.5 High-latitude flux transport and polar filaments
Solar filaments (called prominences when observed at the limb) are long-lived, relatively cool and dense structures suspended by the magnetic field at heights of order 50 Mm above the solar surface (e.g., Mackay et al., 2010). Observations of filaments at high latitudes are, along with polar faculae, the longest available data set providing information on solar magnetism at high latitudes. Filaments and filament channels delineate neutral lines, so the overall structure of the magnetic field can be reconstructed from such data (McIntosh, 1972; Durrant and Wilson, 2003).
McIntosh (2003) presented synoptic charts of the major large-scale magnetic features of the low solar atmosphere, showing the relationship between neutral lines, filaments and coronal holes. Filaments lie along the main neutral lines, and the filaments form a chain-like pattern at high latitudes in each hemisphere, called the polar crown. Gaps in the polar crown are often associated with coronal holes, either extensions of the polar coronal hole or separate high-latitude coronal holes. Filaments and coronal holes are always spatially separated because filaments lie along neutral lines whereas coronal holes are confined to unipolar regions. A secondary crown of filaments sometimes forms at lower latitudes, separating the leading-polartiy active region flux from the trailing-polarity flux, just as the primary polar crown separates the trailing-polarity flux from the polar flux.
The example studied dates from 1982, towards the end of the cycle 21 maximum and just after the polar field had reversed. The main filament had length approximately a solar radius and it lasted about a year. The main activity complex, indicated by the boxes in Figures 48 and 49, lasted about 20 rotations, and the poleward surge of magnetic flux is visible in the magnetic butterfly diagram as a major positive streak meeting the negative polar cap. This surge of flux weakened as it traveled poleward, and it does not appear to have significantly changed the polar field strength. Some reasons for this can be suggested. Although a large quantity of flux was involved (the supercluster of two complexes contained about 1023 Mx at its height), the complexes were arranged in a complicated quadrupolar configuration with many internal neutral lines. As individual bipoles within and between the complexes expanded, fragmented and cancelled, it was mainly the flux at the outer edges of the cluster that survived to be transported poleward. Thus the various filament channels that quickly formed along neutral lines within the cluster did not survive longer than two rotations. After three rotations the flux streaming poleward from the supercluster was arranged in a simple bipolar pattern that exhibited longitudinal stretching due to differential rotation, clearly evident in Figure 48. A lengthy and long-lasting filament channel formed along the neutral line separating the two opposite-polarity bands of flux in the shape of a “switchback”, marked by V-shaped wedges in Figure 49, with the upper arm forming along the neutral line between the positive-polarity flux and the negative polar cap.
Besides delineating frontiers between bodies of opposite-polarity magnetic flux, filaments also mark where twist is trapped in the magnetic field low in the solar atmosphere. This twist can be characterized in terms of the handedness, or chirality, of the field. Filaments are characterized as having dextral or sinistral chirality (handedness) according to the direction of the filament’s axial field relative to the underlying photospheric polarity distribution. A significant preference for dextral prominences to form in the northern hemisphere and sinistral in the south has been observed at both active and high latitudes (Leroy et al., 1983; Martin et al., 1994). In active regions, the main polarity inversion lines (PILs) have chirality consistent with the overall helicity of the active region (Rust and Martin, 1994), where dextral/sinistral chirality corresponds to negative/positive helicity, but the chirality outside active regions is determined by the complex interactions between active regions. Differential rotation tends to add chirality of the “wrong” sign to neutral lines (Leroy, 1978), and small-scale flux emergence and motions are not expected to add significant net chirality of either sign. If a neutral line is oriented north-south then rotational shearing can create chirality of the “correct” sign (DeVore, 2000), but most neutral lines created by decayed active region flux is highly slanted as in, e.g., Figure 48, making the creation of the observed high-latitude chirality patterns by high-latitude rotational shearing unlikely. Therefore, it appears that chirality of the correct sign must be transported from active to high latitudes. The hemispheric rule is evidently caused by the emergence of active regions whose helicity sign satisfies the rule, not the poleward transport of the active region flux, which merely redistributes the helicity within the same hemisphere (Wang et al., 2013).
The NSO He i 10830 A spectrohelio grams and Ottawa River Solar Observatory Hα images analyzed by Gaizauskas et al. (2001) indicated that significant quantities of negative magnetic helicity accumulated where the active regions in the main complex emerged, and this helicity was still present in the poleward-migrating flux after the complexes had disappeared. Since emergence of opposite-polarity fluxes with negative helicity leads naturally to dextral filaments in the northern hemisphere, consistent with the hemispheric chirality rule described by Martin et al. (1994), 10 filament channels associated with the initial flux emergence and dispersal and its subsequent poleward transport all shared dominant dextral chirality. The physical cause of the hemispheric filament chirality rule may be related to the poleward flux transport of the decayed, helicity-carrying active region fields that form the filaments. The magnetic helicity is introduced into the atmosphere when the active regions emerge, and is carried to high latitudes in the decayed active region flux. The relative helicity of the highly conducting magnetic fields of the solar atmosphere is approximately conserved on the time scales of interest (Berger, 1984), and so when active region flux decays, is transported poleward, and meets flux of opposite polarity, it is expected that the helicity will accumulate along the neutral line.
4.6 Prominence eruptions and coronal mass ejections
We next discuss the fate of the filaments when they reach high latitudes at times of polar field reversal. Figures 47 and 50 indicate that these high-latitude crowns of filaments disappear during the period of polar reversal. Signatures of polar field reversal include the disappearance and reforming of polar coronal holes as well as the disappearance of polar filaments following their rush to the poles. These two processes are related by the effect of open flux being replaced at the poles with flux of opposite polarity. Before open flux of one polarity is replaced by open flux of opposite polarity, the intervening neutral lines must be removed, a process signaled by the eruption of the filaments that lie along them. Low (2001) has argued that since relative magnetic helicity is approximately conserved in solar atmospheric fields (Berger, 1984), the high-latitude filaments can only be removed by ejection of the twisted field. The filaments’ disappearance is therefore a necessary consequence of the polar reversal.
The high-latitude prominence eruptions and coronal mass ejections (CMEs) provide a natural mechanism for removing relative magnetic helicity and for the disappearance of polar crown filaments that participate in the rush to the poles. Because eruptive prominences are almost always accompanied by CMEs (Munro et al., 1979), prominence eruptions and CMEs can often be identified with each other (Gopalswamy et al., 2003b). We will explore the solar cycle patterns of prominence eruptions and coronal mass ejections (CMEs) in this section.
Gopalswamy et al. (2003b) studied the statistics of the association rate, relative timing and spatial relationships between prominence eruptions and CMEs. Among their statistical sample, 72% (134) of the prominence eruptions were associated with CMEs. While the prominence eruptions and CMEs began around the same time, with no cycle-dependence in their temporal relationship (Gopalswamy et al., 2003b), the position angle offset between associated prominence eruptions (observed off-limb near the solar surface) and CMEs (observed in the LASCO field of view) shows a clear cycle-dependence when plotted on a butterfly diagram as in Figure 51 (bottom, Gopalswamy et al., 2012). During cycle minimum the prominence eruptions occur at systematically higher latitudes than the locations where the CMEs appear. Figure 51 indicates a significant average positive offset (prominence eruptions were more poleward than coronal mass ejections) when the polar coronal holes were most prominent, i.e., during solar minimum. This position angle offset is believed to be due to strong polar fields. The intervals of exclusively positive offsets span the solar minima, from the beginning of the SoHO/LASCO data series in 1996 until early 1998, and from early 2007 until late 2010. This average positive offset disappeared whenever significant activity appeared, when the influence of the polar coronal holes on the eruptions appears to have been reduced. Although the polar field strength was comparatively weak during the cycle 23/24 minimum the average offset angle, 19°, was about the same as that during the cycle 22/23 minimum, 22°. Few events were recorded during each minimum.
Luhmann et al. (2011) suggested that the observed enhanced CME rate of cycle 24 may be connected to the weak polar fields allowing more ejections to escape into the heliosphere. Petrie (2013) analyzed the CME rates recorded in the Computer Aided CME Tracking (CACTus Robbrecht et al., 2009) and Solar Eruptive Event Detection System (SEEDS Olmedo et al., 2008) catalogs, both based on SOHO/LASCO coronagraph data, and found evidence that the CME rate itself may be dependent on the polar field strength. Whereas the CME rate as measured from coronagraph data collected by numerous satellites flown during cycle 21 and the rise of cycle 22 (1975–1989) was very well correlated with the sunspot number (Webb and Howard, 1994), the CACTUS and SEEDS CME rates were much less well correlated with the sunspot number over cycle 23 and the rise of cycle 24 (1997–2012). In particular, the ratio of CME rate per sunspot number was systematically higher after the cycle 23 polar reversal than before. Wang and Colaninno (2014) countered that the change of cadence of the LASCO images in 2010 may have been responsible for the increase in CME detection. Normalizing the CME rate using the assumption that the CME detection rate is proportional to the image rate, they found a much higher correlation between the normalized CME rate and the sunspot number and concluded that the polar field strength could have no more than a second-order effect on the CME rate. Petrie (2015) found that the Coordinated Data Analysis Workshops (CDAW) and CACTus LASCO CME rates for CMEs of angular width > 30°, both based on LASCO/C2 and C3 images, matched each other closely, and sharply increased (per sunspot number) on completion of the cycle 23 polar field reversal, around 2004. The SEEDS CME rate, based on C2 images alone, differed from the CDAW and CACTus rates in increasing (per sunspot number) only on the onset of cycle 24, in 2010. It remains to be seen which conclusion is more accurate. It is very difficult to reproduce both the initiation of a CME and its progress through a realistic coronal medium, and so a comprehensive study of the relationship between the polar fields and eruptions has not yet been carried out.
Recently, Gopalswamy et al. (2015) reported that the halo CME rate has been higher during cycle 24 than during cycle 23 (Gopalswamy et al., 2015), and that the distribution of CME sources in apparent longitude has also been much flatter, with proportionally twice as many halo CMEs originating from central meridian distances ≥ 60°. Their explanation is based on the decrease in total (magnetic + plasma) pressure in the corona and heliosphere allowing enhanced CME expansion. Gopalswamy et al. (2014) found evidence for this in a study of CME widths and velocities, whose constant of proportionality has changed by 50% for early cycle 24 compared to early cycle 23. This conclusion, if confirmed, would link the enhanced cycle 24 CME rate to the weakened polar fields via the reduced radial IMF (see Section 4.3).
An impressive and coherent body of knowledge of the polar fields has accumulated from several decades of observational and theoretical work. It has become increasingly evident that the polar fields play a central role in the solar cycle, in the solar interior and atmosphere. The pioneering observations from the Hinode spacecraft have changed our observational view of these fields, from a diffuse collection of weak, nearly unipolar magnetic features, closer to our view of diffuse low-latitude quiet-Sun or coronal-hole fields: of a highly complex and non-uniform mixture of intense, nearly vertical fields and smaller patches of nearly horizontal field. It is the vertical patches that represent the polar fields’ pivotal role in the solar cycle and their global influence, even though the patches of nearly horizontal field are much more numerous. This influence of the vertical fields is seen in the cycle-dependent distribution of the coronal holes, solar wind and ejecta.
Since these fields were first observed in the 1950s, we have learnt that they form from a variety of processes that act in concert in a complex and beautiful way, and that they may form the seed field of the solar cycle that follows, though a final dynamo theory of this phenomenon has not yet been established. The properties of the fields depend not only on the combined behavior of these flows but on systematic biases and asymmetries in the bipolar active regions. These properties have so far been only partially explained by the theory of the bipoles’ buoyant emergence from the interior (Fan, 2009). Even the formation of the polar fields from the active regions’ decay, the part of the cycle that occurs in plain sight, is only imperfectly understood by us, as our recent efforts to explain the weak polar fields of cycle 23 have shown. Many ingredients of the kinematic flux transport models are not well determined by observations. The meridional flow profile measurements tend to differ according to whether magnetic feature tracking, Doppler or helioseismic measurements are used, and the model results are highly sensitive to the details. Likewise, analyses of bipole tilt angles provide essential initial conditions for flux transport calculations, but the results are still noisy and inconclusive. Latitude centroid calculations for active region fields seem to produce more stable results that correlate well with observed polar field changes.
Future knowledge of the polar fields will come from a variety of different sources from many areas of solar physics. Continuous high-resolution vector images of the polar fields, such as the unprecedented high-resolution, multi-level observations from the ground-based Daniel K. Inouye Solar Telescope (DKIST), will reveal the detailed behavior of the fields over the cycle. The physical nature of the flux-cancellation processes behind active region decay and polar field reversal can only be revealed by such observations. They will also give us an improved estimate of the polar magnetic flux into the atmosphere and its changes over time, crucial information for the study of the polar fields’ great influence in the heliosphere. At the most basic level, observations of the polar fields are hampered by the large viewing angle from our observing position in the ecliptic plane. It would take an out-of-ecliptic satellite, carrying a good magnetograph, to overcome this limitation. The proposed Polarimetric and Helioseismic Imager (PHI) on ESA’s Solar Orbiter may address this problem. More realistically, vector synoptic magnetograms covering the full solar surface are a highly desirable data product, one that would have application in global modeling of the solar interior and atmospheric magnetic fields. Several obstacles lie in our path. Besides the regions of the solar surface that cannot be observed from Earth for significant periods of time, it is difficult to obtain reliable Stokes parameters from the weak fields that dominate polar latitudes for, although there are kilogauss fields at the poles, they are difficult to resolve on a routine basis because of their small size and the large viewing angle from the ecliptic plane.
The magnetic conditions in the chromosphere much more closely match the conditions in the corona, and so a further desirable data product is a map of the chromospheric field. The SO-LIS/VSM already provides line-of-sight full-disk magnetograms and synoptic magnetograms for the chromospheric field, and a chromospheric vector magnetograph is under development. Because the chromospheric field is, unlike the photospheric field, not approximately radial, a vector measurement is necessary to estimate the chromospheric flux in the atmosphere without relying on annual averages based on viewing angle changes associated with the B0-angle. But, again, it remains to be seen how far poleward reliable chromospheric vector field measurements can be made on a routine basis. A practical compromise is to combine the easier-to-obtain line-of-sight measurements for weak fields, including high latitudes, with full-Stokes vector data for the active regions.
Surface flux transport modeling with magnetogram data assimilation is becoming an essential part of the effort to construct the most accurate possible snapshot of the global photospheric flux distribution at any time. The flux transport model parameters, in particular the meridional flow profile amplitude and shape and the diffusion/dispersal rate, are not tightly constrained by observations. The meridional flow may change significantly in time and may have different effects on fields of different strength, as may also the magnetoconvective coupling. Further detailed observational information is needed to constrain the models and control their behavior. The global atmospheric models extrapolated from improved magnetograms, derived from an optimal combination of observations and modeling, will give us improved estimates of coronal hole locations and structure and solar wind speed distributions, which will enable better forecasts of space weather events, including CME propagation.
Kinematic flux transport and dynamo models have played an essential role in helping us to understand the cyclical behavior of the polar fields, but at present they are only kinematic. Increasingly sophisticated numerical models, and guidance from helioseismology, have advanced the modeling of dynamical processes in the interior, including the interior flows that connect the active regions and the polar fields (Miesch, 2005). Helioseismic measurements of interior flows are telling us more and more about the large-scale flow patterns inside the Sun that, according to the models, must transport old polar flux to lower latitudes and amplify its strength. It is still not known how and at what depth the dynamo process(es) shear the fields and strengthen them, whether near the surface, at the tachocline or somewhere else.
The solar field is behaving in ways unfamiliar to us from previous cycles. The ongoing polar reversal will doubtless prompt new studies on this topic, just as the recent weakness of the cycle 23 polar fields did. The activity level is significantly lower, the polar fields much weaker and north-south asymmetry in the active regions and polar fields has become more common. The consequences of these patterns for the future of both the polar and active region fields give us an excellent opportunity to learn much about the physics of these fields.
Complementing the latest high-resolution observations and detailed physical models, continuous, long-term full-disk magnetogram time series, such as those from NSO and WSO, are essential resources for the long-term study of the polar fields, since only they capture their global behavior on their time-scales of evolution, which are measured in years and cycles. One major message of this review is that, alongside new and exciting high-resolution telescopes, the long-running synoptic projects must be scrupulously maintained if we are to continue to develop our knowledge of the global solar field, including the polar fields.
An error in the tilt angle analysis of Dasi-Espuig et al. (2010) was pointed out by Ivanov (2012), and corrected by Dasi-Espuig et al. (2013), where an anti-correlation remained in the normalized Kodaikanal sunspot tilt angles for cycles 15–21 that they analyzed, and in weakened form in the MWO data. McClintock and Norton (2013) confirmed the existence of this correlation in the MWO data overall, but found that there was no correlation in the northern hemisphere, only in the southern hemisphere.
I thank the two referees for their detailed reports, which corrected many errors and made the review more precise in many places. It has been a privilege to review the work of some of the best solar physicists and astronomers. My interest in the Sun’s polar fields developed from some behind-the-scenes programmatic work on the GONG synoptic magnetograms within a team of scientists and programmers, including Jack Harvey, Frank Hill, Richard Clark and Tom Wentzel. I thank these colleagues, and Luca Bertello, Carl Henney, Aimee Norton, Alexei Pevtsov and Valentín Martínez Pillet for various discussions. Many colleagues, notably Janet Luhmann, Giuliana DeToma and Nick Arge, have been very supportive of our work in this field, for which I am very grateful.